Charge Inversion by Flexible Polyelectrolytes on Flat Surfaces from

Macromolecules 2005, 38, 8911-8922

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Charge Inversion by Flexible Polyelectrolytes on Flat Surfaces from Self-Consistent Field Calculations Qiang Wang† Department of Chemical and Biological Engineering, Colorado State University, Fort Collins, Colorado 80523-1370 Received May 8, 2005; Revised Manuscript Received August 16, 2005

ABSTRACT: We have applied a continuum self-consistent field (SCF) theory to flexible polyelectrolytes on flat surfaces either uncharged or carrying charges opposite to the polyelectrolytes. We examined in detail the effects of various parameters on the polyelectrolyte adsorption and surface charge compensation by the adsorbed polyelectrolytes. The ground-state dominance approximation (GSDA) was used to explore the large parameter space involved, including the charge distribution and degree of ionization of the polyelectrolytes, surface charge density, short-range (non-Coulombic) surface-polymer interactions, solvent quality, and bulk polymer and salt concentrations. The numerical results under GSDA were also compared with full SCF calculations to examine the effects of molecular weight of the polyelectrolytes. Strong charge inversion is found for relatively long polyelectrolytes on oppositely charged, attractive surfaces in poor solvent at high salt concentrations. At the mean-field level, the adsorption behavior of polyelectrolytes at high salt concentrations can be understood by that of neutral polymers in good solvent.

1. Introduction Theoretical study on polyelectrolyte adsorption has been carried out for about 30 years by now.1,2 In this paper, we limit our discussion to equilibrium behavior of (untethered) flexible polyelectrolyte chains on flat and homogeneous surfaces. Two types of charge distribution on the chain are considered: In the case of “smeared” charge distribution, each polymer segment carries the same amount of charge; this is a model for strongly dissociating polyelectrolytes. In the case of “annealed” charge distribution, each polymer segment has the same probability of carrying a charge; this is a model for weakly dissociating polyelectrolytes. The early work of Hesselink assumed a step-function for the polymer segmental density profile in the adsorbed layer.3,4 A better approach is to determine the profile without such a priori assumption; this leads to the self-consistent field (SCF) theory. Both lattice and continuum SCF theories have been used to study polyelectrolytes on or between flat surfaces. These are mean-field approaches, where the only variation in the system is along the direction perpendicular to the surface(s); the polymer chains are also modeled as intrinsically flexible (in addition to electrostatic stiffening effects). Van der Schee and co-workers first extended the lattice SCF theory developed by Scheutjens and Fleer for neutral polymers5,6 to the adsorption of polyelectrolytes having smeared charge distribution, where a uniform dielectric constant was used.7,8 Due to some computational problem, most of their results were actually obtained based on an extension of Roe’s theory.9 Evers et al. extended this formalism further to the adsorption of polyelectrolytes having annealed charge distribution, and considered the case of position-dependent dielectric constant, which was taken as a linear combination (weighted by volume fractions) of those of pure polymer and solvent.10 Using a multi-Stern-layer model, Bo¨hmer et al. considered the finite volume of ions, which was set to be the same as that of the solvent †

E-mail: [email protected].

molecules and polymer segments, and also included nonCoulombic interactions between ions and other species (including surface sites), which were described by the Flory-Huggins χ parameters.11 These authors extended the theory of Scheutjens and Fleer,5,6,12,13 and studied polyelectrolytes confined between two parallel surfaces.11 Their formalism was used by van de Steeg et al., who studied in detail the effects of added salt on the adsorption of polyelectrolytes having smeared charge distribution, and they classified two regimes: the screening-enhanced regime where the adsorbed amount of polymers increases with increasing salt concentration and the screening-reduced regime where the opposite occurs.14 Also using this formalism for polyelectrolytes having smeared charge distribution, Dahlgren and Leermakers examined the depletion zones (where the polymer segmental density is lower than that in the bulk) and the polydispersity effects on polyelectrolyte adsorption.15 Linse systematically investigated the effects of various parameters in this formalism on the adsorption of weakly charged (and strongly dissociating) polyelectrolytes in a Θ-solvent.16 On the basis of the lattice SCF theory, Fleer proposed an analytical “onelayer” model for adsorption of polyelectrolytes in dilute solutions of a good solvent on uncharged surfaces; good agreement with the numerical SCF results was obtained for highly charged polyelectrolytes.17 In all the above studies, either a constant charge density or a constant electrostatic potential was used for the surface(s); this is in general not the case in experimental systems. Vermeer et al. introduced a dissociation equilibrium for surface sites, so that the charges carried by the surface can be controlled by solution pH.18 In addition to this, Shubin and Linse used the site-binding model to account for the binding equilibrium between dissociated surface sites and cationic species (salt cations and dissociated polymer segments) in the first layer closest to the surface.19 The lattice SCF studies have greatly furthered our understanding on the behavior of polyelectrolytes on/ between flat surfaces. The use of a underlying lattice,

10.1021/ma050960b CCC: $30.25 © 2005 American Chemical Society Published on Web 09/21/2005

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however, is an artifact that reduces the numerical accuracy in these calculations. This also makes it difficult to deal with the cases where the polymer segments, solvent molecules, and ions have different sizes. We therefore choose the continuum SCF theory. The governing equations in most previous mean-field studies in continuum20-37 can be derived from our SCF equations presented in section 2 under various approximations. The ground-state dominance approximation (GSDA) has been commonly used to derive or solve the governing equations in these studies, with a few exceptions.21-23,31,32 In addition, we note that, under GSDA, Joanny combined the scaling arguments for the electrostatic blob model with mean-field equations to study the polyelectrolyte adsorption in Θ-solvent.38 A similar method was also used by Dobrynin, who studied the effects of solvent quality on polyelectrolyte adsorption.39 Also under GSDA, Huang and Ruckenstein included explicitly the short-range (van der Waals) interactions between the surface and polymers in their governing equations (not through boundary conditions as shown below).40 Through various analytical and numerical results, these continuum mean-field studies have provided us with important insights on polyelectrolytes on/between flat surfaces. Recent study on polyelectrolyte adsorption has focused on the surface charge overcompensation (or charge inversion) by the adsorbed polyelectrolytes37-39,41-46 due to its essential role in the polyelectrolyte layer-by-layer assembly process.47-49 This phenomenon has been observed in experiments.50-56 Here we define r as the ratio of the compensated surface charge (by the adsorbed polyelectrolytes) to the surface charge (including signs). In lattice SCF calculations, although only slight charge inversion (r ≈ 0) was reported in earlier work,11,14,16,18 Shubin and Linse reported strong charge inversion (r < -1) for polyelectrolytes with a small degree of ionization (0.034) on an attractive (through non-Coulombic interactions with the polymers) surface at high salt concentrations in Θ-solvent.19 By analytically solving the mean-field equations, Joanny predicted full/strong charge inversion (r e -1) for indifferent/attractive surfaces at high salt concentrations in Θ-solvent.38 His results, however, were questioned very recently by Shafir and Andelman, who numerically solved the same equations without the approximations made by Joanny; they found only moderate charge inversion (-1 < r < 0) for attractive surfaces at high salt concentrations in Θ-solvent.37 In this work, we present numerical results of the continuum SCF theory for flexible polyelectrolytes on flat surfaces. Our theoretical formalism and numerical methods are presented in section 2. In section 3, we examine in detail the effects of various parameters on the surface charge compensation by the adsorbed polyelectrolytes, including the charge distribution and degree of ionization of the polyelectrolytes, surface charge density, short-range (non-Coulombic) surface-polymer interactions, solvent quality, and bulk polymer and salt concentrations; GSDA is used to explore the large parameter space involved. Strong charge inversion is found for polyelectrolytes on oppositely charged, indifferent or attractive surfaces in poor solvent at high salt concentrations. In section 4, we follow the work of Joanny38 and analytically solve the mean-field equations under GSDA; this allows us to understand the adsorption behavior of polyelectrolytes at high salt concentra-

Macromolecules, Vol. 38, No. 21, 2005

tions from that of neutral polymers in good solvent. The analytical results are also quantitatively compared with the numerical results to assess its applicability. In section 5, we compare the numerical results under GSDA with full SCF calculations to examine the effects of molecular weight of the polyelectrolytes; the strong charge inversion found at high salt concentrations is retained by relatively long polyelectrolytes. The last section is devoted to conclusions. 2. Theoretical Formalism The general formalism of the continuum SCF theory for polyelectrolyte systems has been derived in refs 57 and 58. In this paper, we consider a system of charged homopolymers A with NA segments on each chain in a solution of small-molecule solvent S. We assume that both polymer segments and solvent molecules have the same density F0, and ignore the volume of small ions. We also assume that all ions carrying the same type of charge are identical, and denote cations by + and anions by -. Integer variables υ+ > 0, υ- < 0, and υA are used to denote the increments of unit charge e carried by cations, anions, and polymer segments, respectively; without loss of generality, we set υA e 0 hereafter. The system is in contact with a flat and homogeneous surface at x ) 0; we assume that the system becomes effectively invariant at x ) L . 0, where it is in contact with a bulk solution. The surface is either uncharged or positively charged. We assume that the system is invariant in the directions parallel to the surface, and only consider variations along the direction perpendicular to the surface (denoted by x). 2.1. Self-Consistent Field Equations. As in our previous work,58 we include the following contribution to the Hamiltonian of the system: chain connectivity (treated with the Gaussian chain model), short-range (e.g., van der Waals) interactions between polymer segments and solvent molecules (modeled with the Flory-Huggins χ parameter), and the Coulombic interactions between all charges in the system (excluding the surface charges, which are considered later through boundary conditions). We then perform the HubbardStratonovich transformation that introduces into the canonical partition function of the system the electrostatic potential ψ(x) (in units of kBT/e, where kB is the Boltzmann constant and T the absolute temperature), a conjugate field ωj(x) for each species j ()A, S, +, -), and a pressure field that imposes the incompressibility constraint everywhere in the system (note that all these fields are pure imaginary in our notation). Finally, the following set of SCF equations can be obtained under the saddle-point approximation

φA(x) )

φ h AN QANA

(

∫0N /N ds q(x, s)q x, A

)

NA -s N

ωS(x) ) ωA(x) + χN[2φA(x) - 1] F1(x) ≡ φA(x) +

[

]

φ hS ωS(x) exp -1)0 QS N

(1) (2) (3)

h+ dg(x) υ+φ exp[- v+ψ(x)] + F2(x) ≡ - φA(x) + Q dψ(x) + 2 hυ -φ d ψ(x) exp[- υ-ψ(x)] + ) 0 (4) QN dx2


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where φA(x) denotes the normalized (by F0) polymer segmental density field, which is constrained to the h A ≡ nANA/ corresponding microscopic density by ωA(x); φ h M ≡ nM/(F0V), where nj is the number of (F0V) and φ molecules of species j in the system, V is the volume of the system, and M represents only the small-molecule species (i.e., S, +, and -); N is an (arbitrary) chain length chosen for the normalization discussed below; q(x, s) corresponds to the probability of finding the endsegment of an A polymer of length sN at x, and it satisfies the modified diffusion equation

∂q ∂2q - [ωA(x) - Ng(x)]q ) ∂s ∂x2

(5)

with the initial condition q(x, s ) 0) ) 1, where s ∈[0, NA/N] is a variable along the chain contour; g(x) ≡ -pυAψ(x) for the smeared charge distribution where each polymer segment carries charge pυAe, and g(x) ≡ ln[1 - p + p exp(- υAψ(x))] for the annealed charge distribution where each segment has a probability p of carrying charge υAe. The single-chain/particle partition functions are defined as

QA ≡ QS ≡ Q( ≡

1 l

1 l

1 l

( )

∫0l dx q x,

[

∫0l dx exp -

NA N

]

ωS(x) N

∫0l dx exp[- υ( ψ(x)]

(6)

υApφA,b φ h+ ) cs,b Q+ υ+

(7)

υ+cs,b φ h)Qυ-

(8)

where l ≡ L/Rg, Rg ≡ a xN/6, and a denotes the statistical segment length of polymers; is the dielectric constant, which is assumed to be uniform for x g 0. Note that to obtain the above equations we have made the following normalization: x/Rgf x and 3kBT/(2πF0e2a2) f (where before the normalization is in units of 4π0 with 0 ) 8.854 × 10-12 (A‚s)2/(J‚m) being the permittivity of vacuum). Once the SCF equations are solved, the mean-field free energy (of mixing) per polymer segment or solvent molecule can be calculated as (in units of kBT)

f)

1

[

l dx χ[1 - φA(x)]2 ∫ 0 l

ωA(x)

( )] dψ

2

N 2N dx φ h A QA QM ln - φ h M ln (9) NA φ hA φ hM M -

∑

Readers are referred to ref 58 for details on the above derivation. Finally, the incompressibility and the charge neutrality constraints on the whole system give

φ hS ) 1 - φ hA υApφ h A + υ+φ h + + υ-φ h- +

σSF lxN

)0

proportional to the short-range (van der Waals) interactions between the surface and polymers. For surfaces attracting the polymers, d-1 < 0, and vice versa. The d-1 ) 0 case is referred to as an indifferent surface, which corresponds to the boundary condition of (∂q/ ∂x)|x)0 ) 0. The limit of d-1 f ∞ is referred to as a nonadsorbing surface, which corresponds to the boundary condition of q(x ) 0, s) ) 0.59 On the other hand, the surface can be held either at a constant electrostatic potential, which gives ψ(x ) 0) ) ψSF (where the surface charge density is solved from the SCF equations), or at a constant surface charge density, which gives (dψ/dx)|x)0 ) -σSFxN/ (where the surface electrostatic potential is solved); note that there is no difference between the two solutions for the corresponding cases. At x ) l, we use (∂q/∂x)|x)l ) 0 and (dψ/dx)|x)l ) 0 as boundary conditions. We set the salt concentration cs,b (normalized by F0) and the polymer segmental density φA,b in the bulk solution. We also use the bulk solution as the reference state for the electrostatic potential where we set ψ ) 0; the Poisson-Boltzmann distribution of ions (refer to the second and third terms on the left-hand-side of eq 4) therefore gives

(10)

where the normalized surface charge density σSF ≡ σ0x6/(F0a), and σ0 denotes the number of surface charges (in units of e) per unit area. We set σSF g 0 in this paper. 2.2. Boundary Conditions and Numerical Solution. At x ) 0, we use the following boundary condition for q(x, s): [q - d (∂q/∂x)]|x)0 ) 0, where d-1 is

where we have assumed that one salt molecule generates one cation. This effectively couples our system with the bulk solution for ions. To couple our system with the bulk solution for polyelectrolytes, we solve for φ hA such that

F3 ≡ φA(l) - φA,b ) 0

(11)

In general, we solve the modified diffusion equation by the Douglas scheme,60 except for the case of indifferent surfaces where a pseudo-spectral method61 with fast cosine transforms is used. The Douglas scheme is a finite-difference method superior to the commonly used Crank-Nicolson scheme in that it provides higher accuracy in the spatial domain with little extra work,60 while the pseudo-spectral method has even higher accuracy in both the spatial and temporal (s) domains61 (but is limited to certain boundary conditions). In both methods, the temporal domain [0, NA/N] is discretized (uniformly) into nNA/N subintervals (where N is chosen such that nNA/N is an integer). The SCF equations are solved in real space. The spatial domain [0, l] is discretized (uniformly) at m + 1 nodes; we denote the position of node i by xi ) il/m, where i ) 0, ..., m. We choose X ≡ {{ωA(xi)}, {ψ(xi)}, φ h A} as independent variables (σSF, instead of ψ(x0), is chosen for cases of constant surface electrostatic potential), and F ≡ {{F1(xi)}, {F2(xi)}, F3} ) 0 as corresponding equations.62 Second-order-accurate finite differences are used to numerically evaluate the spatial derivatives of ψ(x), i.e., (dψ/dx)|x)xi ) [ψ(xi+1) - ψ(xi-1)]m/(2l) and (d2ψ/dx2)|x)xi ) [ψ(xi+1) - 2ψ(xi) + ψ(xi-1)]m2/l2, where ψ(xm+1) ) ψ(xm-1) and ψ(x-1) ) ψ(x1) + 2lxNσSF/(m), consistent with the boundary conditions. A composite Simpson’s rule is used to evaluate all integrals. As noted in ref 58, {ωA(xi)} in the SCF solution can be uniformly shifted by an arbitrary constant; we

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therefore set QS ) φ h S to obtain unique {ωA(xi)}. The Broyden’s method combined with a globally convergent strategy63 is used to solve the set of nonlinear equations F(X) ) 0, which provides fast convergence and high accuracy. In this work, the maximum absolute value of the residual errors of these equations is less than 10-10. In addition to the residual errors of F(X), there are other numerical errors introduced by using finite values of m, n, and l. We choose l to be large enough so that both |φA(l) - φA,b| and |ψ(l)| are less than 10-10. Our values of n are on the order of 103-104, with larger n used for larger m or NA. Because of the presence of the surface, the numerical accuracy is in general dictated by the discretization of the spatial domain; large values of m are needed to resolve the sharp variations near the surface. We use the following practical criterion to choose m: the absolute value of the difference between h A and that from QS calculated according to QS ) 1 - φ eq 7 is less than 10-10; this typically requires m be on the order of 103. 2.3. Ground-State Dominance Approximation. ∞ The q(x, s) in eq 5 can be expanded as q(x, s) ) ∑k)0 exp(- ωks)qk(x), where ωk and qk(x) are the eigenvalues and eigenfunctions of -d2/dx2 + ωA(x) - Ng(x), respectively. For an infinitely long chain (NA f ∞) in the “bound” state where the differences between the smallest eigenvalue ω0 and the others are finite, the groundstate dominance approximation can be used to truncate this expansion after the leading term.64 This reduces h A/QA and eq 5 to eq 1 to φA(x) ) exp(-ω0NA/N)q02(x)φ d2q0/dx2 ) [ωA(x) - Ng(x) - ω0]q0(x), from which one obtains

F4(x) ≡ d2 xφA(x) - [ωA(x) - Ng(x) - ω0] xφA(x) ) 0 (12) dx2 The SCF equations can therefore be reduced to two coupled ordinary differential equations (ODEs), eqs 12 and 4, for xφA(x) and ψ(x); ωA(x) in eq 12 is calculated as

|φA(l) - φA,b| and |ψ(l)| are less than 10-10; this requires that L be from 84a to 2100a in our calculations, depending on the system parameters. Same as in the above full SCF calculations, the numerical accuracy is generally dictated by the discretization of the space domain. For cases of charged polymers, we extrapolate our numerical results to m f ∞, based on the observation that all the calculated quantities are linear with l/m for m J 104.66 Once the SCF equations are solved, the mean-field free energy (of mixing) per polymer segment or solvent molecule can be calculated as (in units of kBT)

f)

1 l

dψ ∫0l dx[χφA2(x) + ln[1 - φA(x)] - 2N ( dx ) ] + 2

Q+ Qφ h Aω 0 -φ h + ln -φ h - ln (15) N φ h φ h +

-

2.4. Calculated Quantities. In this work we focus on two calculated quantities: One is the amount of adsorbed polymers, Γ ≡ xN ∫l0 dx [φA(x) - φA,b]; polymers are depleted from the surface when Γ < 0, and adsorbed when Γ > 0. The other is the surface charge compensated by the adsorbed polymers, σ ≡ σSF + vA xN∫l0 dx [pA(x)φA(x) - pφA,b], where pA(x) ≡ - dg(x)/(υA dψ(x)) is the local degree of ionization of the polymers; for charged surfaces, we have exact charge compensation when σ ) 0, charge inversion when σ < 0, and full charge inversion when r ≡ σ/σSF ) - 1.67 For neutral polymers, eq 12 can be solved analytically under the approximation ln[(1 - φA,b)/(1 - φA(x))] ≈ φA(x) - φA,b; for l f ∞, this gives

Γ)

x

(d-1)2 Nυ

2

+

2φA,b d-1 υ xNυ

x

2φA,b υ

(16)

where υ ≡ 1 - 2χ is the dimensionless excluded volume parameter. On the other hand, by integrating eq 4 with our boundary conditions, we get

ωA(x) ) χN[1 - 2φA(x)] - N ln[1 - φA(x)] (13)

∫0l dx [exp(- υ+ψ) - 1] + l υ+cs,b xN ∫0 dx [exp(- υ-ψ) - exp(- υ+ψ)]

h A. where we have set QS ) 1 - φ At x ) 0, we use the following boundary condition for xφA(x)

Exact charge compensation is therefore always obtained when one sets both φA,b ) 0 and cs,b ) 0.

|

d xφA xφA dx 1

) d-1

(14)

x)0

We also use (dxφA/dx)|x)l ) 0. The rest of the boundary conditions are the same as before; this gives ω0 ) χN(1 - 2φA,b) - N ln(1 - φA,b). Our equations are identical to those used by Andelman and co-workers,33-37 if we use F0 ) a-3, χ ) (1 υF0)/2 (where v is the excluded-volume parameter used in their work), and the approximation

ln

1 - φA,b 1 - φA(x)

≈ φA(x) - φA,b

The ODEs are solved by a relaxation method;65 where a convergence criterion of 10-15 (defined in ref 65) is used. Again, we choose l to be large enough so that both

σ ) υApφA,b xN

3. Numerical Results under GSDA Hereafter, we only consider monovalent systems and set υA ) υ- ) - υ+ ) - 1. In this section, we use the ground-state dominance approximation (GSDA) to explore the large parameter space involved in our model for flexible polyelectrolytes on flat surfaces, including the charge distribution of the polyelectrolytes (smeared vs annealed), degree of ionization of the polyelectrolytes p (varied from 0 to 1), surface charge density σSF (varied from 0 to 0.2), short-range (non-Coulombic) surfacepolymer interactions characterized by d-1 (varied from -10 to ∞), solvent quality characterized by χ (varied from 0 to 1), bulk polymer segmental density φA,b (varied from 10-5 to 10-3), and bulk salt concentration cs,b (varied from 0 to 0.1). For a ) 0.5 nm and F0 ) a-3, φA,b ) 1.25 × 10-4 corresponds to a bulk polymer segmental density of about 1.66 mM, cs,b ) 0.1 corresponds to a bulk salt concentration of about 1.33 M, and σSF ) 0.1 corresponds to a surface charge density of about 2.61 ×



Figure 2. Comparison of polymer segmental density profiles φA(x) between the smeared and annealed charge distributions with the degree of ionization p ) 0.1. Also shown is the local degree of ionization pA(x) for the annealed case. The results are obtained under the ground-state dominance approximation and for nonadsorbing surfaces (d-1 f ∞), with σSF ) 0.1, cs,b ) 0, χ ) 0, and φA,b ) 1.25 × 10-4.

Figure 1. Effects of degree of ionization p, bulk polymer segmental density φA,b, surface charge density σSF, and solvent quality χ on the surface charge compensation σ for nonadsorbing surfaces (d-1 f ∞) in salt-free cases (cs,b ) 0). The results are obtained under the ground-state dominance approximation and for the smeared charge distribution. σSF ) 0.1 and χ ) 0 in part a, and p ) 0.5 and φA,b ) 1.25 × 10-4 in part b. The filled circle in part b corresponds to an uncharged surface (σSF ) 0) in a good solvent (χ ) 0).

10-2 C/m2. At T ) 300 K, ψ ) 1 corresponds to an electrostatic potential of about 25.9 mV. Under these conditions, a dielectric constant of 80 corresponds to ≈ 0.343 (which is used in this paper). 3.1. Nonadsorbing Surfaces. Here we set d-1 f ∞. Figure 1 shows the effects of degree of ionization p, bulk polymer segmental density φA,b, surface charge density σSF, and solvent quality χ on the surface charge compensation σ in salt-free cases for the smeared charge distribution. We see that σ in most cases is negligible compared to σSF, except when σSF j 10-4 (where no charge inversion is obtained). Such small values of σ require accurate calculations of Γ (thus large values of m and l); in all of our numerical results, σ has at least three significant figures. The almost exact charge compensation leads to Γ ≈ σSF/p for the smeared charge distribution where pA(x) ) p. This implies that φA,b and χ, the parameters representing short-range interactions in the system, have little effects on Γ when the electrostatic interactions dominate; this is supported by our data (not shown). The same conclusion can be drawn on the effects of d-1, the parameter characterizing the short-range surface-polymer interactions (refer to Figure 6 below). As expected, for too small surface charge density (σSF j 2 × 10-5 for p ) 0.5, φA,b ) 1.25 × 10-4, and χ ) 0), the

effects of the infinitely large d-1 become dominant: polymers are depleted from the surface due to the loss of chain conformational entropy near the surface and Γ becomes negative. The difference between the smeared and annealed charge distributions depends on both ψ(x) and p; it diminishes when ψ(x) ) 0, p ) 0, or p ) 1. In Figure 2, we compare the polymer segmental density profiles φA(x) for these two charge distributions, and also show the local degree of ionization pA(x) for the annealed case; other conditions (σSF ) 0.1, p ) 0.1, cs,b ) 0, χ ) 0, and φA,b ) 1.25 × 10-4) are the same for these two cases, which give ψSF ≈ 2.69 in the smeared case and 1.88 in the annealed case. Here we see that polyelectrolytes in the annealed case carry much more charges in the vicinity of the surface but have smaller φA(x) from the surface to approximately the location of the first local minimum of φA(x). This gives no significant difference in the surface charge compensation: σ ≈ 9.34 × 10-5 in the smeared case, and 4.68 × 10-5 in the annealed case. The difference between these two charge distributions becomes smaller for smaller |ψ(x)| (e.g., as the bulk salt concentration cs,b increases). Hereafter, we focus on the smeared charge distribution, where the adsorbed amount and charge compensation are simply related through σ ) σSF - pΓ. The above results are for salt-free cases. Consistent with the previous numerical SCF study in continuum,36 Γ decreases (σ increases) with increasing bulk salt concentration cs,b (not shown); i.e., we are in the screening-reduced regime.14 At high enough cs,b, electrostatic interactions are largely screened, and the polymers are depleted from the surface,36 again due to the dominating effects of the infinitely large d-1. For nonadsorbing surfaces, therefore, only slight charge inversion (if any) can be obtained. 3.2. Moderate Charge Inversion on Indifferent Surfaces in Θ-Solvent. Here we set d-1 ) 0 and χ ) 1/ . This case was studied analytically by Joanny, who 2 predicted full charge inversion at high salt concentrations from the mean-field equations.38 Figure 3 shows the effects of cs,b, σSF, φA,b, and p on the surface charge compensation σ. We do see that the charge inversion, or Γ equivalently, increases with increasing cs,b, i.e., we are in the screening-enhanced regime;14 but only moderate charge inversion ( -1 < σ/σSF < 0) was obtained

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Figure 4. Decay of ψ(x) to 0 becomes very slow as φA,b approaches 0. The results are obtained under the ground-state dominance approximation and for the smeared charge distribution, with d-1 ) 0, σSF ) 0.1, p ) 0.5, cs,b ) 0.1, and χ ) 1/2. All curves virtually overlap in the inset.

Figure 3. Effects of bulk salt concentration cs,b, surface charge density σSF, bulk polymer segmental density φA,b, and degree of ionization p on the surface charge compensation σ for indifferent surfaces (d-1 ) 0) in Θ-solvent (χ ) 1/2). The results are obtained under the ground-state dominance approximation and for the smeared charge distribution. p ) 0.5 and φA,b ) 1.25 × 10-4 in part a, and σSF ) 0.1 and cs,b ) 0.1 in part b. Dotted curves are from the analytical theory, and the others are numerical results. The circles in part a correspond to saltfree cases (cs,b ) 0) with σSF ) 0.1: the filled one is the numerical result and the open one is from the analytical theory. In part b, σ/σSF from the analytical theory for φA,b ) 1.25 × 10-4 slightly increases with decreasing p; while it should be 1 for neutral polymers (p ) 0), the analytical theory gives a limit of about -0.9318 as pf 0.

from the numerical results within reasonable parameter range. It is also interesting to see a minimum in σ/σSF as σSF or p varies. Our results agree with the recent work by Shafir and Andelman, who numerically solved eqs 4 and 12 with φA,b ) 0 and Θ-solvent under the approximation ln[(1 - φA,b)/(1-φA(x))] ≈ φA(x) - φA,b; they found only moderate charge inversion for attractive surfaces (-6 e d-1 j - 0.462) at high salt concentrations.37 Under the same conditions, their charge inversion is only slightly larger than our results, since φA(x) is small in these calculations. A common problem associated with numerical calculations at φA,b ≈ 0 is demonstrated in Figure 4: even at a high salt concentration of cs,b ) 0.1, the decay of ψ(x) to 0 becomes very slow as φA,b approaches 0; this requires large values of l in highaccuracy calculations. We therefore use 10-5 as the smallest φA,b in our calculations. 3.3. Strong Charge Inversion at High Salt Concentrations. The above results show that σ ≈ 0 when the electrostatic interactions in the system dominate;

Figure 5. Effects of solvent quality χ on the amount of adsorbed polymers Γ and the surface charge compensation σ for an indifferent surface (d-1 ) 0). The results are obtained under the ground-state dominance approximation and for the smeared charge distribution, with σSF ) 0.1, p ) 0.5, cs,b ) 0.1, and φA,b ) 1.25 × 10-4. Dotted curves are from the analytical theory, and the others are numerical results.

high salt concentrations are therefore necessary for obtaining at least moderate charge inversion (within the mean-field theory). Under such conditions, the solvent quality and short-range surface-polymer interactions also play important roles in obtaining strong charge inversion. Figure 5 shows the effects of solvent quality, characterized by χ, on Γ and σ for an indifferent surface (d-1 ) 0) at a high salt concentration (cs,b ) 0.1); almost 600% charge inversion is obtained at χ ) 1 from the numerical results! Note that this condition is not unusual because water is a poor solvent for hydrophobic polymer backbones. Such polyelectrolytes are watersoluble because of the large entropy gain of dissociated counterions and electrostatic interactions in the system, both of which are described by other terms in our theory, and not by the solvent quality characterized by the χ parameter. We note that poor solvent conditions have rarely been considered in previous mean-field studies on polyelectrolyte adsorption.7,8,10,18,31,39 The effects of the short-range surface-polymer interactions, characterized by d-1, on Γ are shown in Figure 6. We see that d-1 has small or negligible effects when the electrostatic interactions in the system dominate, i.e., for polyelectrolytes in salt-free cases; this leads to σ ≈ 0 and thus Γ ≈ σSF/p discussed above. However,



Figure 6. Effects of d-1, the parameter characterizing the short-range surface-polymer interactions, on the amount of adsorbed polymers Γ. The results are obtained under the ground-state dominance approximation and for the smeared charge distribution. σSF ) 0.01, p ) 0.5, and χ ) 1 in the cases of polyelectrolytes (PE) at cs,b ) 0.05; χ ) 0 in all other cases; and φA,b ) 1.25 × 10-4 in all the cases. Dotted curve is from the analytical theory, and the others are numerical results. The filled symbols on the right axis are numerical results for nonadsorbing surfaces: the square is for the σSF ) 0.1 and p ) 0.5 case, the triangle is for the σSF ) 0 and p ) 0.1 case, the star is for the case of PE at cs,b ) 0.05, and the circle is for the neutral case (where p ) 0 and σSF ) 0).

d-1 has strong effects for polyelectrolytes at a high salt concentration (cs,b ) 0.05): σ/σSF ≈ - 13.35 at d-1 ) -5. As shown in the figure, the adsorption behavior in this case is strikingly similar to that of neutral polymers in a good solvent (χ ) 0); this will be further investigated in section 4. Figure 7 shows the effects of bulk polymer segmental density φA,b, surface charge density σSF, bulk salt concentration cs,b, and degree of ionization p on the surface charge compensation σ in a poor solvent (χ ) 1). Strong charge inversion (σ/σSF < -1) is found in most cases. Again, we see a minimum in σ/σSF as σSF varies. Note that in Figure 7b p J 0.316 is needed in order to avoid (macroscopic) phase separation of the bulk solution; this is also required by the analytical theory, as discussed below. 4. Analytical Results at High Salt Concentrations To better understand the adsorption behavior of polyelectrolytes at high salt concentrations, where moderate or strong charge inversion is found, we extend the work by Joanny, who analytically solved the meanfield equations at high salt concentrations with φA,b ) 0 and Θ-solvent under GSDA.38 Here we consider only the smeared charge distribution and take l f ∞. We assume ln[(1 - φA,b)/(1 - φA(x))] ≈ φA(x) - φA,b, and |ψ(x)| , 1 at high salt concentrations. Under further approximations, the coupled ODEs, eqs 4 and 12, can be analytically solved (see Appendix for details, where the validity of these approximations are also examined); the final result is given by eq 16, with v and d-1 replaced by υeff ≡ υ + p2/cb and deff-1 ≡ d-1 - xNpσSF/cb, respectively, where cb ≡ pφA,b + 2cs,b. We therefore see, at high salt concentrations, that charged polymers effectively increase the excluded volume parameter, and that a surface carrying opposite charges to the polyelectrolytes effectively makes the short-range surfacepolymer interactions more attractive. It is worth to note a few things about this analytical theory: First, when φA,b > 0, Γ has the opposite sign to

Figure 7. Effects of bulk polymer segmental density φA,b, surface charge density σSF, bulk salt concentration cs,b, and degree of ionization p on the surface charge compensation σ for indifferent surfaces (d-1 ) 0) in a poor solvent (χ ) 1). The results are obtained under the ground-state dominance approximation and for the smeared charge distribution. p ) 0.5 and cs,b ) 0.05 in part a, and σSF ) 0.01 and φA,b ) 1.25 × 10-4 in part b. Dotted curves are from the analytical theory, and the others are numerical results.

deff-1, as evident from eq 16; any charge inversion therefore requires deff-1 < 0. Second, the theory requires

υeff > 0, which leads to p > xυ2φA,b2/4-2vcs,b - vφA,b/2 when υ < 0; this corresponds to p J0.316 in Figure 7b. Finally, for indifferent surfaces in Θ-solvent with any positive values of σSF, p, and cs,b, the analytical theory predicts full charge inversion only when φA,b ) 0; any φA,b > 0 will lead to less charge inversion.68 Predictions of the analytical theory are compared with numerical results in Figures 3, 5, 6, and 7; large deviations from the numerical results are seen in some cases, due to the approximations used in obtaining the analytical results (see Appendix). As expected, the analytical theory works better at higher salt concentrations; it also works better for smaller σSF and χ. Note that it is unable to capture the minima in σ/σSF revealed by numerical results. In all the cases, it overestimates the charge inversion, i.e., σA < σN (when σN < 0), where the superscript “A” denotes the analytical results and “N” the numerical results. Since (σA - σN)/σN ) (ΓA ΓN)/[ΓN(1 - σSF/pΓN)], the analytical results have smaller relative error in the adsorbed amount than in the charge inversion when σN ()σSF - pΓN) < 0. Although the analytical theory can only be used at small σSF (j0.01), it qualitatively explains the mechanism of charge inversion at high salt concentrations, where the electrostatic interactions, although screened,

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Figure 8. Qualitative comparison with Figure 6a in ref 19 on strong charge inversion for polyelectrolytes on attractive surfaces in Θ-solvent at a high salt concentration (cs,b ) 0.1 M). The results are obtained using the analytical theory for p ) 0.034 and φA,b ) 5 × 10-5. See text for details on the comparison.

still play important roles in the system. Taking as an example the polyelectrolytes (p ) 0.5) on an oppositely charged (σSF ) 0.01), indifferent (d-1 ) 0) surface in a poor solvent (υ ) - 1) at a high salt concentration (cs,b ) 0.05), if we ignore the small bulk polymer segmental density (φA,b ) 0), we obtain υeff ) 1.5 and deff-1 ) -0.05xN. The strong charge inversion in this case, σ/σSF ≈ - 2.333, is therefore mainly due to the effectively attractive surface, i.e., the electrostatic contribution to deff-1; the electrostatic contribution to υeff in fact acts against the charge inversion (reduces Γ), and it is counteracted by the poor solvent to keep the charge inversion from being otherwise smaller. For quantitative comparison, if we use φA,b ) 1.25 × 10-4, eq 16 gives σ/σSF ≈ - 1.810, while the numerical solution under GSDA gives σ/σSF ≈ - 1.534 (shown later in Figure 10). In Figure 8, we use the analytical theory to confirm the lattice SCF results of Shubin and Linse, who reported at high salt concentrations strong charge inversion for polyelectrolytes on attractive surfaces in Θ-solvent.19 In their work, the polyelectrolytes consisted of neutral and charged (strongly dissociating) segments with the chain structure mapped from their experimental systems;19 here we focus on the linear chain with the p ) 0.034 case and use the smeared charge distribution instead. Since their chain length is large (NA ) 10 150), GSDA provides a good description in this case (refer to Figure 10 below). Their bulk salt concentration of 0.1 M corresponds to cs,b ≈ 5.49 × 10-3 here (with a ) 0.45 nm used in their work and F0 ) a-3), and we use φA,b ) 5 × 10-5 and Θ-solvent as in their work. Moreover, they used a charge-regulating surface whose charge density varies with solution pH, salt concentration, and the presence of polyelectrolytes; from Figure 2 of their paper, σSF ≈ 0.012 C/m2 (which corresponds to about 0.037 here) at pH ) 6 and cs,b ) 0.1 M with the polyelectrolytes adsorbed, and it decreases with decreasing pH, where charge inversion becomes stronger.19 Here we use surfaces at constant charge density and vary σSF from 0.004 to 0.04. We also vary d-1 from 0 to -6; in their work the short-range surface-polymer attractions were characterized by a negative χ parameter between the surface and neutral polymer segments, and this is further complicated by the finite volume of ions who compete with polymers for surface adsorption. Despite all these differences, we see that the analytical theory well captures the strong charge inversion shown

Figure 9. Comparisons of polymer segmental density profiles φA(x) from the ground-state dominance approximation with those from full self-consistent field calculations, for a uncharged, nonadsorbing surface (σSF ) 0 and d-1 f ∞) in a good solvent (χ ) 0). Key: (a) neutral polymers; (b) polyelectrolytes with no salt (cs,b ) 0); (c) polyelectrolytes (p ) 0.5) with added salt, where the bulk salt concentrations are shown for a ) 0.5 nm and F0 ) a-3. Smeared charge distribution is used for polyelectrolytes, and φA,b ) 1.25 × 10-4 in all the cases.

in Figure 6a of their paper. Note that it is essential to have an attractive surface in this case, since for indifferent surfaces in Θ-solvent only moderate charge inversion can be obtained, as shown in section 3.2. 5. Comparisons between SCF and GSDA Results It is known that GSDA works better for polymers on surfaces or in confined geometries where the chains are in the “bound” state. Here we quantitatively assess the validity of GSDA by comparing its results with full SCF calculations. Since GSDA corresponds to NA f ∞, the comparison also reveals the effects of molecular weight of the polymers. Figure 9 shows the polymer segmental density profiles on a uncharged, nonadsorbing surface (σSF ) 0 and


Figure 10. Comparison of polymer segmental density profiles φA(x) from the ground-state dominance approximation with those from full self-consistent field calculations, for a charged, indifferent surface (σSF ) 0.01 and d-1 ) 0) in a poor solvent (χ ) 1) at a high salt concentration (cs,b ) 0.05). Smeared charge distribution is used with p ) 0.5, and φA,b ) 1.25 × 10-4.

d-1 f ∞) in a good solvent (χ ) 0). We see that for neutral polymers GSDA is good only for rather long chains (NA J 100 000). But it becomes much better for polyelectrolytes. For example, for the case of NA ) 1000 and p ) 0.5 with no salt, the density profile from GSDA virtually overlaps with that from full SCF calculation, except around the peaks. This is due to the presence of counterions in the depletion zone right next to the surface, which attract polyelectrolytes to be closer to the surface than neutral polymers (as seen in the figure); this was examined in detail by Dahlgren and Leermakers using lattice SCF theory.15 Even with added salt, which screens electrostatic interactions, the agreement between GSDA results and full SCF calculations are still improved relative to neutral cases. Note that the uncharged, nonadsorbing surface used here is the worst for polymer adsorption; the polymers are actually depleted (Γ < 0) in all the cases shown in Figure 9. For oppositely charged, less repulsive surfaces that favor the adsorption of polyelectrolytes, we would expect that, in most cases, GSDA results be in quantitative agreement with full SCF calculations. In other words, the molecular weight of the polyelectrolytes does not play a critical role here. To check GSDA results on strong charge inversion, which are obtained at high salt concentrations, we compare in Figure 10 the polymer segmental density profiles on an oppositely charged, indifferent surface (σSF ) 0.01 and d-1 ) 0) in a poor solvent (χ ) 1) at a high salt concentration (cs,b ) 0.05). We see that the strong charge inversion is retained by relatively long chains (NA J 10 000). Finally, since a poor solvent is used here, we check the possibility of (macroscopic) phase separation of the bulk solutions:69 For p ) 0.5 and cs,b ) 0.05, the smallest critical value of χ is 1.25, corresponding to NA f ∞, and it increases with decreasing NA; the bulk solutions used in Figure 10 are therefore stable against (macroscopic) phase separation. We also used the random phase approximation58 to check the possibility of microphase separation of the bulk solutions;70 it cannot occur either in this case. 6. Conclusions In this work, we have applied a continuum selfconsistent field (SCF) theory to flexible polyelectrolytes


on flat surfaces; the surface is either uncharged or carrying opposite charges to the polyelectrolytes. We examined in detail the effects of various parameters on the amount of polyelectrolyte adsorption (Γ) and surface charge compensation by the adsorbed polyelectrolytes (σ). The ground-state dominance approximation (GSDA) was used to explore the large parameter space involved, including the charge distribution and degree of ionization (p) of the polyelectrolytes, surface charge density (σSF), short-range (non-Coulombic) surface-polymer interactions (d-1), solvent quality (χ), bulk polymer segmental density (φA,b), and bulk salt concentration (cs,b). The numerical results under GSDA were also compared with full SCF calculations to examine the effects of molecular weight of the polyelectrolytes. To better understand the adsorption behavior of polyelectrolytes at high salt concentrations, an analytical theory was developed and compared with numerical results under GSDA. When the electrostatic interactions in the system dominate (e.g., when cs,b is small), only σ ≈ 0 can be obtained, which leads to Γ ≈ σSF/p for the smeared charge distribution; parameters representing shortrange interactions in the system (d-1, χ, and φA,b) have only small or negligible effects on Γ. Although the polymer segmental density profiles are different, both the smeared and annealed charge distributions give similar surface charge compensation. At high salt concentrations, d-1 and χ play critical roles in determining the adsorption behavior of the system. For nonadsorbing surfaces (d-1 f ∞), polymers are depleted due to the short-range repulsions between the surface and polymers. For indifferent surfaces (d-1 ) 0) in Θ-solvent (χ ) 1/2), only moderate charge inversion (-1 < σ/σSF < 0) is found within reasonable parameter range. Strong charge inversion (σ/σSF < -1) is obtained by either decreasing d-1 (attractive surfaces) or increasing χ (poor solvent). The electrostatic interactions, although screened in this case, still play important roles in the system: charged polymers effectively increase the solvent quality (decrease χ), and a surface carrying opposite charges to the polyelectrolytes effectively makes the short-range surface-polymer interactions more attractive (decreases d-1). The adsorption behavior of polyelectrolytes at high salt concentrations can therefore be understood by that of neutral polymers in good solvent, as shown by the analytical theory. The molecular weight plays minor role in polyelectrolyte adsorption even at high salt concentrations. There are some limitations in this work: First, we ignored the volume and non-Coulombic interactions of small ions in the system. These ions compete with polymers for surface adsorption and could decrease the surface charge compensation by the polymers. Second, we used a uniform dielectric constant. Since the polymer concentration is low, the dielectric constant in the system can be approximated as uniform. However, the dielectric constant of the surface is in general much lower than that of water, and image charge effects71-73 act against the adsorption of polyelectrolytes. Finally, since our study is at the mean-field level, we ignored correlations and fluctuations in the system, which have been shown to play important roles for polyelectrolytes on/between flat surfaces.41-46,72-81 The necklace formation of polyelectrolytes in poor solvent and its influence on the adsorption are also not captured in our model. These could be topics for future publications.

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Appendix The starting point of the analytical theory is the following coupled ODEs

d2ψ ) (2cs,b + pφA,b)ψ(x) + p(φA(x) - φA,b) N dx2 d2 xφA ) N[υ(φA(x) - φA,b) - pψ(x)]xφA(x) dx2

(17)

σSF

xcb

( x )

Ncb p - (φA(x) - φA,b) cb

exp - x

(18)

(19)

where cb ≡ pφA,b + 2cs,b. Equation 18 then becomes

d2 xφA ) Nυeff(φA(x) - φA,b) xφA(x) dx2 Ncb NpσSF exp -x xcb

( x )x

φA(x) (20)

where υeff ≡ υ + p2/cb. Again, its exact analytical solution (for x ∈[0, ∞)) is unknown. Instead, we solve it in the interval [h,∞) (where h > 0) by ignoring its last term; this requires the second approximation, exp(-hxNcb/) ≈ 0. In addition, a boundary condition for xφA(x) at x ) h is needed; this is obtained by integrating d(d(ln xφA)/dx) ) [(d2xφA/d2x)/xφA - (d(ln xφA)/dx)2]dx from x ) 0 to h using eq 20, which gives

|

1 d xφA dx φ

x

A

) deff-1 + ∆

x)h

where deff-1 ≡ d-1 - xNpσSF/cb and ∆ ≡ ∫h0 dx[Nυeff(φA - φA,b) - (dxφA/dx)2/φA] + (xNpσSF/cb) exp(- hxNcb/). With the third approximation, ∫h0 dx [Nυeff(φA - φA,b) (dxφA/dx)2/φA] ≈ 0 for small h, the problem can then be solved analytically, in analogy to the adsorption of neutral polymers in a good solvent. Finally, we need the fourth approximation, which changes the lower integration limit for Γ from 0 to h; the final result for Γ is given by eq 16, with υ and d-1 replaced by υeff and deff-1, respectively (this is the analytical results plotted in all the figures). The polymer segmental density at the surface, which is approximated by that at x ) h from the analytical theory, is given by

φA,SF ) φA,b +

{

Γ≈-

x

2φA,b veff Nυeff

φA,b2 for deff-1 > 0, and 2(deff-1)2 σSF p ψSF ≈ + φA,b c xcb b

φA,SF ≈

with the boundary conditions (dψ/dx)|x)0 ) -σSFxN/, (dψ/dx)|xf∞ ) 0, (d lnxφA/dx)|x)0 ) d-1, and (dxφA/dx)|xf∞ ) 0; we also have ψ(x f ∞) ) 0 and φA(x f ∞) ) φA,b. The exact analytical solution is unknown, and the analytical theory gives an approximate solution by decoupling the ODEs in a brute-force way. Mathematically, the analytical theory requires four approximations: First, φA(x) in eq 17 is constant; this gives

ψ(x) )

and ψSF can be obtained by substituting the above result into eq 19. These results, as well as that for Γ, can be further simplified in two limits: When (deff-1)2 . 2NυeffφA,b, they become

(deff-1)2 deff-1 (d -1)2 + 2NυeffφA,b Nυeff Nυeff x eff

{

Γ≈-

2deff-1

xNυeff 2(deff-1)2 for deff-1 < 0 (21) φA,SF ≈ Nυeff σSF 2(deff-1)2 p ψSF ≈ cb Nυeff cb

{

when (deff-1)2 , 2NυeffφA,b, they become

Γ≈-

deff-1

xNveff

+

(deff-1)2

x8φA,bNveff

φA,SF ≈ φA,b - deff-1 ψSF ≈

σSF

xcb

+ deff-1

3/2

x x

2φA,b Nυeff

2φA,b p Nυeff cb

The above four approximations used to derive the analytical theory are “uncontrolled”; while the second approximation favors a large h and the last two favor a small h, the first approximation is probably the most problematic (it is obviously not satisfied by the analytical solution). Physically, this approximation corresponds to the case where the decay of ψ(x) is much faster than that of φA(x), as pointed out by Joanny.38 From Figure 11, which shows the numerical solutions to eqs 4 and 12 (strictly speaking, to examine the validity of the first approximation, one should numerically solve eqs 17 and 18, but the differences caused by the approximations ln[(1 - φA,b)/(1 - φA(x))] ≈ φA(x) - φA,b and |ψ(x)| , 1 should be negligible in these cases), we see that this is the case at high salt concentrations (note the different scales of horizontal axes). φA(x), however, is certainly not constant near the surface (note the logarithmic scale for φA(x)) unless both d-1 ) 0 and σSF ) 0; since solving eq 17 requires integration over the entire range of x, this is probably the major source of error for the analytical theory at large σSF, where the variation of φA(x) near the surface becomes sharp. To examine the second and third approximations, we numerically solve eq 20 using the relaxation method65 (where L ) 420a and m ) 1.4 × 106 are used). Figure 12 shows the values of h (which minimizes ∆) and the corresponding ∆ (divided by -deff-1, which is positive in these calculations) as a function of σSF. We see that ∆ becomes comparable to -deff-1 at large σSF; this is also responsible for the large deviations seen in such cases.


Figure 11. (a) Polymer segmental density profile φA(x), and (b) electrostatic potential ψ(x). The results are obtained under the ground-state dominance approximation and for the smeared charge distribution, with p ) 0.5, cs,b ) 0.05, χ ) 1, and φA,b ) 1.25 × 10-4.

Figure 12. Values of h (which minimizes ∆) and the corresponding ∆ as a function of σSF. The results are obtained from the numerical solution to eq 20, with d-1 ) 0, p ) 0.5, cs,b ) 0.05, and φA,b ) 1.25 × 10-4. Note that deff-1 < 0 in these calculations.

We also see that h is indeed small, which favors the fourth approximation. Acknowledgment. The author thanks Adi Shafir and Prof. David Andelman for helpful discussions and for their preprint of ref 37 before its publication. Financial support for this work was provided by Colorado State University, which is gratefully acknowledged. References and Notes (1) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Cambridge University Press: Cambridge, U.K., 1993. (2) Netz, R. R.; Andelman, D. Phys. Rep. 2003, 380, 1. (3) Hesselink, F. Th. J. Electroanal. Chem. 1972, 37, 317.

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(57) Shi, A. C.; Noolandi, J. Macromol. Theory Simul. 1999, 8, 214. (58) Wang, Q.; Taniguchi, T.; Fredrickson, G. H. J. Phys. Chem. B 2004, 108, 6733, 2005, 109, 9855. (59) For nonadsorbing surfaces the initial condition for q becomes q(x > 0, s ) 0) ) 1, while q(x g 0, s ) 0) ) 1 is applied in all other cases. (60) Mitchell, A. R.; Griffiths, D. F. The Finite Difference Method in Partial Differential Equations; John Wiley & Sons, Inc.: New York, 1980; Chapter 2.4. (61) Tzeremes, G.; Rasmussen, K. O.; Lookman, T.; Saxena, A. Phys. Rev. E 2002, 65, 041806. (62) ωA(x0) and F1(x0) are not needed for nonadsorbing surfaces. (63) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in C: the Art of Scientific Computing, 2nd ed.; Cambridge University Press: Cambridge, U.K., 2002; Chapter 9.7. (64) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (65) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in C: the Art of Scientific Computing, 2nd ed.; Cambridge University Press: Cambridge, U.K., 2002; Chapter 17.3. (66) In particular, the integrated total charge density in the system (including the surface charges) is inversely proportional to m; we use this to choose the values of m (>105) in the extrapolation. (67) In some studies, the charge inversion is identified by an extremum in ψ(x); the integrated total charge from the surface to the location of the extremum (including the surface

Macromolecules, Vol. 38, No. 21, 2005 charges) is 0. Since the charges carried by small ions are also included in this integration, it is different from what we consider here: in our calculations of σ, we take into account only the charges carried by the polymers. (68) This follows directly from the inequality xb2+c2 e b + c (for b > 0 and c g 0), where the equal sign is taken only when c ) 0; here b ≡ - deff-1/(xNυeff) ) σSF/p > 0, and c ≡ x2φA,b/υeff. (69) Warren, P. B. J. Phys. II 1997, 7, 343. (70) Boryu, V. Yu.; Erukhimovich, I. Ya. Sov. Phys. Dokl. 1986, 31, 146. (71) Netz, R. R.; Joanny, J. F. Macromolecules 1999, 32, 9013. (72) Messina, R. Phys. Rev. E 2004, 70, 051802. (73) Cheng, C.-H.; Lai, P. Y. Phys. Rev. E 2004, 70, 061805. (74) Akesson, T.; Woodward, C.; Jonsson, B. J. Chem. Phys. 1989, 91, 2461. (75) Beltran, S.; Hooper, H. H.; Blanch, H. W.; Prausnitz, J. M. Macromolecules 1991, 24, 3178. (76) Kong, C. Y.; Muthukumar, M. J. Chem. Phys. 1998, 109, 1522. (77) Carignano, M. A.; Dan, N. Langmuir 1998, 14, 3475. (78) Yethiraj, A. J. Chem. Phys. 1999, 111, 1797. (79) Dobrynin, A. V.; Deshkovski, A.; Rubinstein, M. Phys. Rev. Lett. 2000, 84, 3101. (80) Podgornik, R.; Dobnikar, J. J. Chem. Phys. 2001, 115, 1951. (81) Jonsson, B.; Broukhno, A.; Forsman, J.; Akesson, T. Langmuir 2003, 19, 9914.

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Charge Inversion by Flexible Polyelectrolytes on Flat Surfaces from

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